Bending Stress Calculator

Bending stress (also called flexural stress) is the internal stress that develops in a beam cross-section due to an applied bending moment. Bending stress varies linearly from zero at the neutral axis to maximum at the extreme fiber. The formula is: sigma = M x y / I, where M is the bending moment (lb-in), y is the distance from the neutral axis to the point of interest (in), and I is the moment of inertia (in^4).

The flexure formula is: sigma = M x c / I = M / S, where M is the applied bending moment, c is the distance from the neutral axis to the extreme fiber (the outermost edge), I is the moment of inertia of the cross-section, and S = I/c is the section modulus. The flexure formula gives the maximum bending stress at the extreme fiber. Stresses are tensile on one side and compressive on the other.

The section modulus S = I / c combines the moment of inertia and the distance to the extreme fiber. It directly determines the maximum bending stress: sigma = M / S. Designing for bending, engineers find the required section modulus: S_required = M / F_b, where F_b is the allowable bending stress. They then select a section from tables with at least this S value. S has units of in^3.

For a simply supported beam with a point load P at midspan: M_max = P x L / 4. For a simply supported beam with uniform load w (lb/ft): M_max = w x L^2 / 8. For a cantilever with point load at tip: M_max = P x L. For a cantilever with uniform load: M_max = w x L^2 / 2. Bending moment diagrams are drawn to show M at every point along the beam.

For structural steel (ASTM A36, Fy = 36 ksi), the AISC Allowable Stress Design (ASD) allowable bending stress is 0.66 x Fy = 0.66 x 36 = 23.76 ksi for compact sections (rounded to 24 ksi). For high-strength steel (ASTM A992, Fy = 50 ksi), the allowable bending stress is 33 ksi. AISC Load and Resistance Factor Design (LRFD) uses a different approach with factored loads and phi factors.